One thing I really enjoyed about this lesson was talking with students about all of the different ways they were working to solve this problem. Some students listed out all of the dates on their page, and worked to mark off the different jobs. Others were able to 'skip count' the dates for each job (June 6, June 12, June 18, etc. for the lawn, for example). Some resourceful students noted that there is a year-long calendar in their student planners, and they marked off the jobs in June/July 2014.

Not only did I enjoy hearing from students about what they were trying, but students were listening to each other and trying new approaches based on what their peers were working on (once kids caught wind of the calendar in the planners, that became a very popular strategy!).

I think it's really powerful to leverage students' thinking, especially when it comes to alternative problem solving strategies. The more students are able to think of multiple pathways to problems, the more likely they'll be to persevere when presented with a novel problem...which is really likely on the SBAC (or PAARC, if that's your state) assessments. I make time as often as I can to ask 'did anyone get to this answer, but solve it in a different way?'

I read the task out loud, for students as listeners. Then, there is a second read of problem, for annotation time (this can be out loud or independent). The goal of this section is to help students make sense of the problem.

The questions in bold below are what I say to students, followed by possible student responses.

What information does the problem give us? The problem tells us that the neighbor wants her lawn watered every 6 days and her dog walked every 4 days. She will pay $5 each time the lawn is watered and $6 every time the dog is walked. She’ll also pay $3 extra when the lawn is watered and the dog is walked on the same day. The jobs are going to be done from June 1 to July 20. How did you annotate? (annotate your copy under the doc camera as students share OR use this as an opportunity to recognize student work that is strong, and show that student’s work under the doc cam) Student responses about annotation will vary, depending on how you’ve set it up in your classroom.

What is the question asking us to find? We have to figure out how much money the neighbor will pay for the jobs.

What information do we not know? We don’t know how many times each job will happen and we don’t know when the jobs will happen on the same day.

Are there any words in this problem that you don’t know? If we don’t figure out the meaning of that word, would you still be able to solve this problem? What do you think this word means? (in this problem, there may not be words that students need help with. This is a good questioning sequence to use often, because it builds the idea that students can still solve problems, even if there are some vocabulary words that are new and unknown)

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Students have 4 minutes to work on their own, and then 8 minutes to work in partners. After the independent time, and before students begin working in pairs, I'll show the work of a student who might be using a more novel strategy compared to his/her peers. I'll ask that student what (s)he's trying to do.

In general, as students are working on more novel problems, I expect them to:

Understand the problem using appropriate annotations

Represent the known and unknown information

Seek a strategy to solve the problem by applying understanding of mathematical connections and rules

Determine a solution and check the reasonableness of the answer using alternate methods and re-contextualizing the solution

As students are working, I circulate around the room and check in with each group. I'm asking:

Tell me about your strategy. What are you working on first? Why?

Why is this strategy helpful? Is there anything about this strategy that is not helpful?

What is it that you’re trying to do?

Can you think of a more efficient way?

What concept are you applying here? (this question is likely to be tricky for kids, until they are used to hearing and responding to it)

Once you finish that, what will you do with the information you find?

How will what you’re doing get you closer to answering the question?

Some of the challenges and misconceptions that might come up as students are working:

Students may incorrectly draw a calendar (or, may spend a ton of time trying to draw a calendar). Consider having templates available if students are struggling with this.

Students’ lists of numbers (if they are listing out all 50 days) may not be organized because of the amount of space needed. Consider having blank paper available for work space.

Students may not determine the bonus money. Have students make an explicit plan before they start to solve.

After 12 minutes of work time (4 independent, 8 in partners), I bring the class back together. Students will have had enough work time to have something down. Not every group may have gotten to the final amount of money earned, but all students should have employed a strategy that allows them to see when the lawn will be watered and when the dog will be walked (I see progress as I circulate as pairs are working)

During the partner exploration time, I determine the students who will present their strategies to the class. I'm looking for one student who used a visual representation of any kind, and one student who used multiples.

The focus of this section is on making a list of multiples for each job, and identifying the common multiples for the bonus money. Although we'll talk about multiple student strategies during this section, I want the class to have a clear idea that a list of multiples is the most efficient strategy for finding common multiples (and one that we'll focus on in the next lesson)

Presentation:

First, I'll display the work from one student who used a visual representation to solve this problem (either a List of dates or a Calendar-type strategy). I'll have students turn and talk with their partner about the work they see, and then the student will present his/her thinking. I'll ask the student why (s)he made the choice to use a calendar, and then will call on other students who also used a visual model to add to the conversation.

Then, I'll display the work of a student who used an organized list of Multiples. In the event that no student used a list of multiples, I will display my own work sample. Again, students turn and talk with their partner about the work they see. I'll ask a student to share what they think the owner of the multiples strategy did, and then ask the owner to add any additional thoughts (s)he might have.

Discussion:

In this part of our conversation, I want students to make connections between the strategies and representations. The questions I ask, with possible student responses, are below.

What are the benefits of this strategy? What are the drawbacks? Student responses will vary. Possible benefits: Using multiples is more efficient. It takes up less space. It’s easier to count up the number of days something happened, because it is in an organized list. We could add other jobs on other days easily. Possible drawbacks: it could be easy to miss a day when both events happen, because you have to check both lists. What’s different about the two strategies we saw? Student responses will vary

Which strategy is more efficient? Why? Students should name that skip counting is more organized and efficient (they may or may not use the word multiples at this point)

Of the two strategies we saw, which would work better if you were figuring out how much you were paid for an entire year? Why? Listing the multiples would be a better strategy, because there are 365 days in a year. It would take a long time to list every number between 1 and 365.

What is the word we can use to describe the numbers (s)he listed – for example, the 6, 12, 18, etc. that represent when the lawn was watered? These are multiples (this would be review for my students)

A multiple of a number is the product of that number and another whole number greater than 0. Think about what we can say about the days that you earn the bonus from the neighbor, using the word multiple? Turn and talk with your neighbor. I have 1-2 students share out what they talked about. Students should identify that the days on which the bonus was paid are multiples of both 4 and 6.

Common multiples are multiples that two or more numbers share. I wrap up the conversation with the key point: To find common multiples of two numbers, list the multiples for each number, then compare the two lists to see which factors are in common. I will build on this idea in our next lesson, and students will practice finding the LCM.

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After our discussion, students have the opportunity to work on an extension of the problem. The constraints have changed slightly, but students are still asked to work with multiples and common multiples to solve.

Before students begin work, I help them to make sense of the problem. I ask them to turn and talk with their partners about what information they will need to use from the first part of this problem, in order to work out this part of the problem. I also have the class make a prediction about whether this second scenario will result in more or less money earned. Students vote with a thumbs up/down, and I call on one of each vote to share the thinking behind the prediction.